After defining the spectral theory of orthogonal polynomials on the unit circle (OPUC) and real line (OPRL), I'll describe Verblunsky's version of Szegö's as a sum rule for OPUC and the Killip-Simon sum rule for OPRL and their spectral consequences. Next I'll explain the original proof of Killip-Simon using representation theorems for meromorphic Herglotz functions. Finally I'll focus on recent work of Gambo, Nagel and Rouault who obtain the sum rules using large deviations for random matrices.

I will give an account of my adventures in the wonderlands of mathematics and cryptography. I'll offer some food for thought on how mathematics can be useful in cryptography, how cryptography can motivate research of mathematical interest, and how mathematicians and cryptographers can learn to play well together. A primary focus will be on the quest to find cryptographically useful multilinear maps. Along the way I will share some thoughts that I hope will be helpful on what I learned about community.

AMS Colloquium Lectures

The theory of nonlinear dispersive equations has seen a spectacular development in the last 35 years. The initial works studied the behavior of special solutions such as solitons/traveling waves. Then, there was a systematic study of the well-posedness theory, using extensively tools from harmonic analysis. The last 25 years have seen a lot of interest in the study of the long-time behavior of solutions, for large data. Issues like blow-up, global existence, scattering and long-time asymptotic behavior have come to the forefront, especially in critical problems. We will concentrate the discussion on the energy critical wave equation, in the focusing case. In the defocusing case, it was shown (1990-2000) that all (large) data in the energy space yield global in time solutions which scatter. The focusing case is very different since one can have finite time blow-up, even for solutions which remain bounded in the energy norm, and solutions which exist for all time and are bounded in the energy norm, but do not scatter, for instance traveling waves. In this lecture I will give an overview of the progress in the last 10 years in the program of obtaining a complete understanding of the global dynamics of solutions which remain bounded in the energy space.

In this second lecture we will concentrate on the classification, in terms of their asymptotic behavior, of radial solutions of the focusing energy critical wave equation in the radial three dimensional case. This will include a discussion of the proof of soliton resolution, for solutions bounded in the energy norm (Duyckaerts-Kenig-Merle, 2013).

In this third lecture we will give an outline of the proof of soliton resolution, along a well-chosen sequence of times, for general (non-radial) solutions of the energy critical wave equation which remain bounded in the energy norm (for dimensions 3,4,5,6). This is work of Duyckaerts-Jia-Kenig-Merle from 2016.

MAA Invited Address

Mathematicians often spend their days thinking about ideas that exist only in their minds. In this talk we'll discuss how to use 3D printing to bring models of those ideas into reality, from start to finish. We'll show how to leverage design software to convert mathematical objects into triangular meshes or voxel representations, and then how those digital representations become code that a 3D printer can understand and implement to create real-world objects. Learn how to get started creating your own mathematical 3D design files, level up your existing design skills, or just enjoy watching the process of turning abstract mathematics into physical plastic. Along the way we'll explore the essential importance of failure, not only in the design process but also in the study of mathematics itself.

Lewis Carroll once published a set of mathematical “pillow problems"; brainteasers designed to give the reader something to ponder before going to sleep. One of the problems was this: what is the probability that a randomly chosen triangle is obtuse? This talk will explore a surprising connection between the shapes of polygons and the Grassmann manifold of 2-planes in real or complex n-space. Using this bridge, a number of interesting problems about random polygons and shapes have recently been solved, including explicit computations of some expectations for geometric properties of random polygons and fast algorithms for polygon sampling. The talk will be accessible to a wide mathematical audience, including students, and will also sketch some connections with biology and polymer science.

AMS Josiah Willard Gibbs Lecture

The quantum laws governing atoms and other tiny objects seem to defy common sense, and information encoded in quantum systems has weird properties that baffle our feeble human minds. John Preskill will explain why he loves quantum entanglement, the elusive feature making quantum information fundamentally different from information in the macroscopic world. By exploiting quantum entanglement, quantum computers should be able to solve otherwise intractable problems, with far-reaching applications to cryptology, materials, and fundamental physical science. Preskill is less weird than a quantum computer, and easier to understand.

MAA Invited Address

From Gauss to Today: Class Numbers and p-torsion in Class Groups of Number Fields

Each number field (finite extension of the rationals) has a positive integer associated to it called the class number, defined to be the cardinality of the class group of the field. Class numbers are important objects that arise naturally in many contexts in number theory: for example, Gauss famously investigated class numbers of quadratic fields, in the context of classifying the representation of integers by binary quadratic forms. Today, many deep open questions remain about the structure of class groups and the growth and divisibility properties of class numbers as fields vary over an appropriate infinite family. This talk will focus on the size of the p-torsion subgroup of the class group: it is conjectured that for any number field and any rational prime p, the p-torsion part of the class group of the field should be very small, in a suitable sense, relative to the discriminant of the field. The talk will survey progress on this open problem, from Gauss to today.

AWM-AMS Noether Lecture

Let $M$ be a symplectic manifold and let $\sigma$ be an antisymplectic involution on $M$. The real locus is the fixed point set of the involution. It is a Lagrangian submanifold. Suppose also $M$ is equipped with the Hamiltonian action of a torus $T$. It is possible to define a compatibility between $T$ and $M$. This set of ideas was introduced in a 1983 paper by Hans Duistermaat.

In this talk I will describe some developments in this field since Duistermaat's foundational paper. My contributions in this area are joint work with Liviu Mare, and (in a separate project) with Nan-Kuo Ho, Khoa Dang Nguyen and Eugene Xia.

Interacting particle transport or kinetic collisional modeling was introduced in the last quarter of the nineteenth century by L. Boltzmann and J.C. Maxwell, independently, giving birth to the area of mathematical Statistical Mechanics and Thermodynamics. These types of evolution models concern a class of non-local, and non-linear integro-differential problems whose rigorous mathematical treatment and approximations are still emerging in comparison to classical non-linear PDE theory. Their applications range from rarefied elastic and inelastic gas dynamics including very low temperature regimes for quantum interactions, collisional plasmas and electron transport in nanostructures, to self-organized or social interacting dynamics. Based on a Markovian framework of birth and death processes, under the regime of molecular chaos propagation, their evolution is described by equations of non-linear collisional Boltzmann type. We will discuss recent progress in analytical and numerical methods covering form initial and boundary value problems, long time dynamics and stability issues.

The mathematical nature of dispersion is the starting point of a very rich mathematical activity that has seen incredible progress in the last twenty years, and that has involved many different branches of mathematics: Fourier and harmonic analysis, analytic number theory, differential and symplectic geometry, dynamical systems and probability. In this talk I will give examples of these diverse directions and related open problems.

MAA Retiring Presidential Address

Mathematics is often valued for its ability to describe the world in beautiful ways. Indeed, beauty is one of many ideals to which we aspire. But why does the practice of mathematics often fall short of our ideals and hopes? How can the deeply human themes that drive us to do mathematics be channeled to build a more beautiful and just world in which all can truly flourish?

AMS Invited Address

Langlands proposed an extraordinary correspondence between representations of Galois groups and automorphic forms, which has deep, and completely unexpected, implications for the study of both objects. The simplest special case is Gauss' law of quadratic reciprocity. In the so called `regular, self-dual' case much progress has been made in the roughly 40 years since Langlands made these conjectures. In this talk I will discuss recent progress in regular, but non-self-dual case. In this case the automorphic forms in question can be realized as cohomology classes for arithmetic locally symmetric spaces, i.e quotients of symmetric spaces by discrete groups. Thus instead of the Langlands correspondence being a relationship between algebra and analysis, it can be thought of as a relationship between algebra and topology. This realization of the Langlands correspondence is in many ways more concrete. It also admits to generalizations not envisioned by Langlands, for instance relating mod p Galois representations with mod p cohomology classes. In this talk I will describe the expected Langlands correspondence in the setting of locally symmetric spaces. I will try both to present the general picture and to give numerical examples. I will also describe recent theorems of Lan, Harris, Thorne and myself on the Langlands correspondence in this setting and startling progress of Peter Scholze in the mod p case. I will not attempt to describe the proofs.

Discrete subgroups of Lie groups play an important role in various areas of mathematics. Lattices, discrete subgroups of finite covolume, are fairly well understood, revealing a dichotomy of flexibility and rigidity. Lattices in $\mathrm{SL}(2,\mathbb{R})$ are flexible, each such lattice has a deformation space of positive dimension, which is closely related to the Teichmüller space of a surface. Lattices in $\mathrm{SL}(n,\mathbb{R})$ with $n\geq 2$ are super-rigid, due to a celebrated theorem of Margulis. It is rather difficult to get a handle on discrete subgroups which are not lattices. I will discuss new developments in geometry, low-dimensional topology, number theory, analysis and representation theory that led to the discovery of several interesting families of discrete subgroups which are not lattices, but - quite surprisingly – admit an interesting structure theory, which arises from a combination of flexibility and rigidity. A particular exciting aspect is the discovery of higher Teichmüller spaces and their relation to various areas in mathematics, such as analysis, algebraic geometry, geometry, dynamics, representation theory.

The difficulties of detecting association and establishing causation have fascinated mankind since time immemorial. Democritus, the Greek philosopher, noted the importance of proving causality when he wrote, “I would rather discover one cause than gain the kingdom of Persia."

To address the difficulties of establishing causation, statisticians have developed many inferential techniques. Perhaps the most well-known method stems from Karl Pearson's correlation coefficient, which was introduced in the late 19th century based on ideas of Francis Galton.

I will introduce in this lecture the recently-devised distance correlation coefficient and describe its advantages over Pearson's and other measures of correlation. We will apply the distance correlation coefficient to data from large astro-physical databases, where it is desired to classify galaxies. The lecture also will analyze data arising from a study of the association between state-by-state homicide rates and the stringency of state gun laws.

The lecture will review a remarkable singular integral arising in the theory of distance correlation coefficients. We will show extensions of this integral to truncated Maclaurin expansions of the cosine function and in the theory of spherical functions on symmetric cones.

What does it mean to be random? We all encounter randomness every day---it is part of how we talk about the weather, sports, and even love. Despite being so familiar, randomness has proven to be an elusive idea to pin down. Even mathematicians have struggled to define randomness, leading to competing and sometimes conflicting definitions. Whatever it is, randomness is a driving force behind many modern computational algorithms. These algorithms --- the Metropolis Algorithm, Markov chain Monte Carlo Methods, and others --- use randomness as the secret ingredient that makes it possible to tackle famously difficult problems such as the Traveling Salesperson Problem and image reconstruction. Using many pictures (and even a few Bob Dylan references), this lecture will reveal the historical quest to define randomness and illustrate how randomness allows us to solve many of today's most challenging applied mathematics problems.

The dynamics of the Einstein equations feature the formation of black holes. The latter are related to the presence of trapped surfaces in the spacetime manifold. The mathematical study of these phenomena has gained momentum since D. Christodoulou's breakthrough result proving that in the regime of pure general relativity trapped surfaces form through the focusing of gravitational waves. (The latter were observed for the first time last year by LIGO.) The proof combines new ideas from geometric analysis and nonlinear partial differential equations (pde) as well as it introduces new methods to solve large data problems. These methods have many applications beyond general relativity. D. Christodoulou's result was generalized by S. Klainerman and I. Rodnianski. In this talk, we investigate the dynamics of the Einstein equations, focusing on these works. Moreover, we address the question of stability of black holes and what has been known so far, involving recent works of many contributors.

In 2015, Karim Adiprasito, June Huh, and Eric Katz announced a proof of a 50-year old conjecture of Heron, Rota, and Welsh asserting that the coefficients of the characteristic polynomial of a matroid form a log-concave sequence. The proof of this conjecture establishes far more: the authors prove combinatorial analogues of Poincare duality, the hard Lefschetz theorem, and the Hodge-Riemann relations for the Chow ring of an arbitrary matroid. This work opens up new horizons in both algebraic geometry and combinatorics, applying deep algebro-geometric intuition to a combinatorial problem with no direct link to algebraic geometry.

The Erdos discrepancy problem asks whether every assignment of signs to the natural numbers must have large imbalances among the multiples of some integer. In 2015 Tao found a remarkable proof that there must be such irregularities. The key is a logarithmic version of the Chowla and Elliott conjectures from multiplicative number theory, which Tao established building upon another breakthrough of Matomaki and Radziwill on multiplicative functions in short intervals. My goal will be to describe some of the story and ideas behind this proof.

Data currently generated in the fields of ecology, medicine, climate science and neuroscience often contain tens of thousands of measured variables. Statistical analyses can result in publications whose results are irreproducible.

The field of modern statistics has had to revisit the standard hypothesis testing paradigm. A first step has consisted in the correction for multiplicity in the number of possible variables selected as significant using multiple hypotheses correction and FDR control (Benjamini, Hochberg, 1995). However this does not solve the problem of post-selection inference where the same data is used to choose models and evaluate them. Recent work by groups at the Wharton school (Brown et al.) and at Stanford (Taylor et al.) address these issues.

It remains that the complexity of software and flexibility of choices in tuning parameters can bias the output towards inflation of significant results; neuroscientists recently revisited the problem in the context of fMRI studies.

Formal correction methods cannot accommodate the flexibility available to today's statisticians. I will also present ways that open source tools (Huber, 2015) and publication of code and data can enhance reproducibility.

AMS Invited Address

Modeling of a wide range of physical phenomena leads to tracking fronts moving with curvature-dependent speed. A particularly natural example is where the speed is the mean curvature. If the movement is monotone inwards, then the arrival time function is the time when the front arrives at a given point. It has long been known that this function satisfies a natural differential equation in a weak sense but one wonders what is the regularity. It turns out that one can completely answer this question. It is always twice differentiable and the second derivative is only continuous in very rigid situations that have a simple geometric description. The proof weaves together analysis and geometry.

Modern biological data present multiple challenges akin to code breaking. We can learn from the master codebreakers who worked during World War II how to leverage patterns using mathematics and statistics. Turing and his fellow codebreakers used graphs and alignments to process their data. I will show how similar approaches provide rich insight into the complex dynamics of the human microbiome. We still use the Bayes factors and diversity indices developed in the 1940's to detect changes in the bacterial communities.

MAA-AMS-SIAM Gerald and Judith Porter Public Lecture

Mathematics for Art Investigation

Mathematical tools for image analysis increasingly play a role in helping art historians and art conservators assess the state of conservation of paintings, and probe into the secrets of their history. The talk will review several case studies, Van Gogh, Gauguin, Van Eyck among others.